Higher Order Derivatives of Laplace Transform/Examples/Example 1

Examples of Use of Laplace Transform of Integral

Let $\laptrans f$ denote the Laplace transform of the real function $f$.

$\laptrans {t^2 e^{2 t} } = \dfrac 2 {\paren {s - 2}^3}$

Proof

\(\ds \paren {-1}^2 \laptrans {t^2 e^{2 t} }\)

\(=\)

\(\ds \dfrac {\d^2} {\d s^2} \laptrans {e^{2 t} }\)

Laplace Transform of Integral

\(\ds \leadsto \ \ \)

\(\ds \laptrans {t^2 e^{2 t} }\)

\(=\)

\(\ds \dfrac {\d^2} {\d s^2} \dfrac 1 {s - 2}\)

Laplace Transform of Exponential

\(\ds \)

\(=\)

\(\ds \dfrac 2 {\paren {s - 2}^3}\)

simplification

$\blacksquare$

Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Important Properties of Laplace Transforms: $7$. Multiplication by $t^n$